## Abstract

We investigate the intrinsic high speed modulation responses of nanolasers and nanoLEDs using bulk, quantum wells (QWs), and quantum dots (QDs) based on a rigorous rate-equation model, which incorporates the optical energy confinement factor to properly account for the negative permittivity and dispersive metal plasma property. We then investigate the dependence of the bandwidth and the energy per bit on the quality factor and the normalized optical volume. We find out that the conditions for the energy per bit less than 50 fJ/bit and 10 fJ/bit are the normalized optical modal volume less than 20 and 5, respectively. In addition, with a uniform quantum dot size in a nanocavity, quantum-dot metal-cavity nanolasers exhibit the largest bandwidth among three types of active materials, and a low energy per bit. With their insensitivity to temperature, quantum-dot metal-cavity nanolasers are favorable for future high speed light sources.

© 2012 Optical Society of America

## 1. Introduction

With the increasing demand of high bandwidth for short-reach applications such as data swapping, processing among computers, chip-to-chip, and intrachip interconnects, current electronics are facing their bandwidth limits [1]. Therefore, using nanolasers and nano-light-emitting-diodes (nanoLEDs) to replace them has been widely investigated for this purpose. The advantages of nanolasers and nanoLEDs lie not only in their low power consumption due to high single-mode spontaneous emission coupling into the cavity mode, but also in their high modulation bandwidth. One of the figures of the merit for a laser is the energy per bit, which is defined as the ratio of the supplied power at which the maximum bandwidth is reached to the maximum bandwidth. Obviously, a smaller value indicates a better energy efficiency. Furthermore, the energy per bit of a laser less than 50 fJ/bit is required for competing with the current electrical interconnects [1]. Recently, 8.76 fJ/bit has been experimentally demonstrated using a buried heterostructure photonic-crystal laser [2]. While the device needs optical pumping, it provides a bandwidth of 20 Gbit/s. Because of the optical pumping, an external modulator is needed for the pumping source to provide a high-speed modulation. The investigation of the electrical injection for the similar device is underway [3]. To evaluate the energy per bit of nanolasers, a rigorous treatment of the rate equations is needed to guide the future direction. For example, the spontaneous emission of nanoLEDs is enhanced by the Purcell effect [4], so the carrier lifetime is shorter and the bandwidth increases. The ultrahigh bandwidth of nanoLEDs has been obtained according to the Purcell factor and oversimplified rate equations [5–8], where the Purcell factor is phenomenologically placed in the spontaneous emission rate without a rigorous treatment, and the material dispersion is not taken into account in the rate equations. In [9, 10], the Purcell factor is placed in the full model and the saturation of *Q* is correctly addressed. In this work, we introduce our rate equations [11] and show that the Purcell effect naturally emerges in the single-mode spontaneous emission rate through Fermi’s golden rule with the consideration of plasmonic dispersion. In addition, we apply the rate equations to two novel metal-cavity nanolasers and nanoLEDs and demonstrate their ultimate bandwidths when the active region consists of bulk, quantum wells, or quantum dots. Metal-cavity lasers have shown the capability to confine light in a subwavelength region [12–15] and are good candidates for nanolasers and nanoLEDs although most metal-cavity lasers show low power at present. Most recently, the advent of a metal-cavity VCSEL with a few *μ*W power [16,17] operating at room temperature paves the way to realize metal-cavity nanolasers and nanoLEDs. In addition, our rate equations with thermal effects show an excellent agreement with the experiment [18]. Therefore, it is an appropriate model to study the thermal effect on the modulation response.

To see the effect of the small optical modal volume on the high-speed modulation response, we employ our proposed metal nanocavity [19] in Fig. 1(a). A p-i-n layer structure for current injection is used with a total thickness of 220 nm, which is around a half effective wavelength (*λ*_{0}/2n), where *λ*_{0} and n are the wavelength in free space and the refractive of the material, to sustain a small cavity. With a thick intrinsic layer, a high optical energy confinement factor can be further obtained. The top and bottom silver layers are used for contact metals, mirrors, as well as heat dissipation. To avoid a short circuit, an insulator layer of SiN* _{x}* is inserted between the sidewall metal and the semiconductor core, and its thickness is optimized for the smallest modal loss. An etch-stop layer of n-InGaAsP layer is used for the substrate removal in the fabrication process. With bonding to a silicon substrate, the device is readily integrated with electronic integrated circuits. In addition, a high single-mode spontaneous emission coupling factor is expected in a small radius because only one dielectric mode (HE

_{111}mode) exists. Thus, it has a small threshold current. Finally, its azimuthal-independent far-field pattern allows it to have a high coupling efficiency for the following interconnects. Due to these advantages, we use this structure to investigate the dependences of the maximum bandwidths of nanolasers and nanoLEDs and the energy per bit of nanolasers on quality factors and optical modal volumes for different gain media. Although some advantages such as a high single-mode spontaneous emission coupling factor disappear in larger volumes, we assume that those properties are still valid throughout in the paper.

On the other hand, we propose a new metal-cavity surface-emitting nanolaser with hybrid mirrors, which is shown in Fig. 1(b). In contrast to the structure in [18], which demonstrated the maximum high power among current metal-cavity lasers and used more than 20 pairs of top and bottom DBRs, only 5.5 pairs of n-doped and p-doped DBRs with a metal layer are employed in this structure to reduce the cavity volume and provide a higher optical energy confinement factor while still sustaining a high reflectivity. The total thickness of the structure is 2 *μ*m and much less than that of the conventional VCSELs, which is tens of *μ*m. In addition, the structure is favorable for quantum well and quantum dot gain media, since the standing wave enhancement factor can further increase the confinement factor and reduce the threshold gain, which is beneficial for high speed modulation. To study the thermal effects on the modulation bandwidth, we apply the quantum well structure in [18] in this new nanolaser and compare it with a quantum dot gain medium. Future improvement and potential of the structure will be addressed.

This paper is organized as follows: First, we introduce the rigorous rate equations with the detailed definition for each term. Second, we use the rate equations to study the modulation bandwidths of nanolasers and nanoLEDs based on the metal-nanocavity structure in [19], taking into account the spontaneous emission coupling into the cavity mode rigorously. The intrinsic maximum bandwidths of nanolasers and nanoLEDs using bulk, quantum wells (QWs), and quantum dots (QDs) are demonstrated. Moreover, the energy per bit of nanolasers for different gain media is evaluated to examine the feasibility of nanolasers applied in optical interconnects. Third, we investigate the thermal effect on the modulation response of QW and QD nanolasers in Fig. 1(b). Finally, we conclude that, with a uniform quantum dot size and three-dimensional confinement of carriers, quantum-dot metal-cavity nanolasers are favorable for future high-speed ultrasmall light sources because of their low energy per bit and insensitivity to temperature.

## 2. Rate equations for nanolasers and nanoLEDs

Although the rate equations have been discussed widely, most of them do not consider the dispersive material and treat the normalization of the optical field properly. In this paper, we introduce the rate equations in Eq. (1) and (2) based on our rigorous derivations [11], taking into account plasmonic dispersion and negative permittivity of the metal plasma. We should point out that these rate equations are applicable to both metal and dielectric cavities, from nano-, to micro-, to macroscale lasers:

*n*= carrier density (cm^{−3})*I*= injection current (A)*η*_{i}= current injection efficiency*q*= electron unit charge (Coulomb)*V*_{a}= active volume (cm^{−3})*R*_{nr}(*n*) = nonradiative recombination rate (cm^{−3}· s^{−1})*R*_{sp}(*n*) = total spontaneous emission rate (cm^{−3}· s^{−1})*R*_{st}(*n*) = stimulated emission coefficient (s^{−1})*S*= photon density (cm^{−3})- Γ
_{E}= optical energy confinement factor *β*_{sp}(*n*) = spontaneous emission coupling factor*τ*_{p}= photon lifetime (ns)

Here, the optical energy confinement factor, Γ_{E}, is used to correctly account for the negative permittivity and dispersive properties of the metal plasma:

*ε*_{R}= the real part of the relative permittivity*V*_{eff}= the effective optical modal volume (cm^{−3})(=*V*_{a}/Γ_{E})*ℰ*(_{m}**r**) = the phasor of the optical electric field

The subscript “a” indicates the active region. The nonradiative recombination rate accounting for the surface recombination and Auger recombination is:

where*v*_{s}= the surface velocity (cm · s^{−1})*A*_{a}= the surface area of the active material (cm^{2})*C*= Auger recombination coefficient (cm^{6}· s^{−1})

The total spontaneous emission rate contains all of the discrete cavity modes and continuous modes:

*τ*

_{sp,rad}[20]. Although there should exist only one cavity mode in a nanocavity, we keep

*m*to distinguish

*R*

_{sp,m}(

*n*) from

*R*

_{sp}(

*n*) and refer to

*R*

_{sp,m}(

*n*) as the single-mode spontaneous emission rate.

Therefore, the single mode spontaneous emission coupling factor in a laser is defined as

Obviously, *β*_{sp}(*n*) also depends on the carrier density *n* and saturates after the threshold condition is reached. A high *β*_{sp}(*n*)*R*_{sp}(*n*) enhances the photon density, *S*, based on the steady-state solution:

On the other hand, the photon lifetime is defined as:

*ω*= the resonant angular frequency of_{m}*m*-th mode (*rad*· s^{−1})*Q*_{abs}= the quality factor due to absorption and scattering loss*Q*_{rad}= the quality factor due to radiation loss

Finally, *R*_{st}(*n*) and *β*_{sp}(*n*)*R*_{sp}(*n*) are expressed as:

- |
*ê*·**p**_{c,v,K}| = the optical momentum matrix element (kg · cm · s^{−1}) - Γ
= the half width at half maximum (HWHM) linewidth of the optical transition energy (eV)_{cv} - Γ
= the half width at half maximum (HWHM) linewidth of the optical mode density (eV)_{c}

*=*

_{c}*h*̄Δ

*ω*=

_{m}*h*̄

*ω*/2

_{m}*Q*. The normalization condition [11] for the electric field in a nanocavity is used for deriving Eq. (11). The detailed derivation for

*β*

_{sp}(

*n*)

*R*

_{sp}(

*n*) is in Appendix 1. A similar procedure is applicable for

*R*

_{st}(

*n*), which can be found in [11].

With a rigorous treatment of the stimulated emission and spontaneous emission, our rate equations are derived for nanolasers and nanoLEDs with the dispersion, and they maintain a similar form as the textbook equations [21, 22]. The importance of our rate equations is the introduction of the optical energy confinement factor to take into account the plasma dispersion and the negative permittivity of metal; therefore, the optical energy is always positive. Moreover, the Purcell effect is automatically included in our single-mode spontaneous emission rate *β*_{sp}(*n*)*R*_{sp}(*n*). For example, to maximize *β*_{sp}(*n*)*R*_{sp}(*n*), a few conditions have to be met: first, the polarization of the optical field has to align with that of the optical dipole direction according to |*ê*·**p**_{c,v,K}|. Second, the resonant frequency of the cavity has to coincide with that of the transition energy. Third, *β*_{sp}(*n*)*R*_{sp}(*n*) is enhanced by a small *V*_{eff}. This addresses the fact that a small *V*_{a} does not necessarily enhance *β*_{sp}(*n*)*R*_{sp}(*n*) if Γ_{E} cannot be sustained at a sufficiently high level. This is why we need a nanocavity: it provides a high Γ_{E} even in a small cavity, and enhances *β*_{sp}(*n*)*R*_{sp}(*n*). Finally, *β*_{sp}(*n*)*R*_{sp}(*n*) can also be enhanced by *Q*, but it saturates when *Q* is extremely high when Γ* _{c}* is much smaller than Γ

*. Different from [5–10], based on our rigorous model,*

_{cv}*Q*is embedded in a Lorentzian function instead of being pulled out from the integral. Our rigorous derivation clarifies the Purcell effect without any ambiguity although an analytical expression, the Purcell factor, is not used. The fact is that the Purcell factor is a simplified expression, which takes the form of the ratio of the total spontaneous emission rate in a cavity to its counterpart in free space based on the corresponding optical mode density. It offers a good physical intuition, but overestimates the effect of

*Q*on the enhancement. Therefore, the bandwidth of nanoLEDs is overestimated when it is directly used in the rate equations [5–8]. Equation (11) can be readily and analytically carried out from [21].

## 3. Dynamics for metal-nanocavity light emitting devices

#### 3.1. Modulation responses

To obtain the high-speed modulation response, we replace the carrier density, *n*, the current, *I*, and the photon density, *S*, in Eq. (2) by *n*_{0} + Δ*n*(*ω*), *I*_{0} + Δ*i*(*ω*), and *S*_{0} + Δ*s*(*ω*), respectively. The subscript “0” indicates the steady-state solution, and the symbols with Δ in front represent small signals. By following a small signal analysis in [21], we obtain the high-speed modulation response function *M*(*ω*) = Δ*s*(*ω*)/Δ*i*(*ω*) and the normalized high-speed modulation response function *M*(*ω*)/*M*(0), which can be expressed in terms of the relaxation angular frequency *ω _{r}*, and the damping factor

*γ*:

*ε*: Other parameters are defined as:

*τ*′

_{sp,Δn}, which is the reciprocal of

*∂*[

*β*

_{sp}(

*n*)

*R*

_{sp}(

*n*)]/

*∂n*, decreases as

*Q*increases, and changes the modulation bandwidth of nanoLEDs. In the following, we study the high-speed modulation bandwidth based on Fig. 1(a) while our results are applicable for all types of nanolasers and nanoLEDs with material and plasmonic dispersions. The detailed design rule for the structure can be found in [19].

First, we study the bandwidth of metal-cavity nanolasers and nanoLEDs with three different gain media. (1) The bulk material is In_{0.53}Ga_{0.47}As. (2) The strain-compensated multiple-quantum-well structure consists of 20 pairs of InGaAsP/InGaAlAs to improve the optical confinement factor, and to provide a high differential gain due to more symmetrical conduction and valence bands from the strain effect, and has shown a broad bandwidth of 30 GHz in an edge emitting laser [23]. (3) For the quantum dots, we refer to the most advanced technology, submonolayer deposition, which grows QDs with a high dot density, N* _{QD}*, and a narrow linewidth [24, 25]. The layer grown by submonolayer deposition usually contains a few stacks of quantum dot sheets with a GaAs spacer inserted in between and these InAs submonolayers are vertically coupled. Based on [24], we assume that each layer has five stacks and each stack consists of 0.5 ML In

*Ga*

_{x}_{1−}

*As and 1.5 ML GaAs, producing a layer thickness of 3 nm. Therefore, we treat each layer as a quantum dot sheet with the dot density as 2×10*

_{x}^{12}cm

^{−2}and assume each equivalent quantum dot with a height of 3 nm and a radius of 2.5 nm due to the high dot density. The inhomogeneous broadening parameter,

*σ*, is set as 5 meV, corresponding to a full width at half maximum (FWHM) as 11.8 meV. The QDs grown by submonolayer deposition have demonstrated impressive performance at 980 nm and 850 nm [26], however, devices working at longer wavelength are desired since they provide a small turn-on voltage, thus, a small power consumption. The preliminary results using In

*Ga*

_{x}_{1−x}As QDs for long wavelength have been demonstrated [26]. Thus, we predict the performance of 1260 nm QDs grown by submonolayer deposition based on the assumption above and consider 3 and 12 quantized energies for conduction and valence bands [27], respectively, for the conservative estimation. The surface recombination lifetime of a quantum-dot laser is modeled as 0.05 ns. The gains of bulk and quantum-well lasers are carried out from four-band Luttinger-Kohn Hamiltonian with the axial approximation to calculate the valence subbband structures [21], and the gain of the QD laser can also be found in [21].

We carry out Γ_{E} and *β*_{sp}(*n*) through the finite-difference time-domain method (FDTD) provided that the gain medium is bulk and the radius R in Fig. 1(a) is 220 nm, and the thicknesses of p-InP, In_{0.53}Ga_{0.47}As, n-InP, n-InGaAsP, and SiN* _{x}* are 30 nm, 160 nm, 30 nm, 20 nm, and 50 nm, respectively. Γ

_{E}is 0.6 for bulk and the values for QW and QD light-emitting devices vary according to the active material volumes and they are 0.2 and 0.044, respectively. Such high Γ

_{E}’s for different gain media result from our metal nanocavity and lead to high speed modulation response.

*β*

_{sp}is 0.41 using the approach in [28]. We assume that it remains the same for other volumes and materials.

*τ*

_{sp,rad}can be deduced from Eq. (7). Notice that

*β*

_{sp}obtained from the FDTD method is bias independent; we assume that the number from the FDTD method is the same as the

*β*

_{sp}when the threshold condition is reached. In addition, R varies from 5 nm to 1500 nm for the investigation of the dependence of the modulation response and the energy per bit on the volume. The parameters used in the calculation are summarized in Table 1.

Based on Table 1, we carried out the maximum bandwidths of the bulk, quantum well, and quantum dots cavities versus *Q* and the normalized optical modal volume V_{n}, defined as *V*_{eff}/(*λ*_{0}/2n)^{3}. The results are shown in Fig. 2.

We discuss bulk and quantum well nanocavities first. In an extremely small volume, e.g., V_{n} = 0.001 in Fig. 2(a) and 2(b), *S*_{0} can be augmented due to an enhanced *β*_{sp}(*n*)*R*_{sp}(*n*) according to Eq. (8) and causes the spectral hole burning. Therefore, the material gain *decreases* with bias significantly. Because of this, even at a high *Q*, the device cannot reach the threshold condition, and operates as a nanoLED. This explains why both bulk and QW semiconductors work as nanoLEDs at an extremely small V_{n}. However, quantum wells have more symmetrical conduction and valence bands, allowing its single-mode spontaneous emission rate to increase with carriers faster than that of the bulk. Therefore, QW nanoLEDs have a higher bandwidth. In addition, due to their high single-mode spontaneous emission rate, QW lasers are more sensitive to spectral hole burning. Thus, they have to work at larger V_{n} around 0.02 and when *Q* is sufficiently high. On the contrary, QDs devices behave as lasers for all conditions in Fig. 2(c) and 2(d) because of the assumption for uniform quantum dot sizes, which allow QDs to sustain a high material gain even though they still suffer spectral hole burning. In addition, QDs have a smaller gain suppression coefficient [29] than those of bulk and quantum wells [22], which alleviates the spectral hole burning further. More details about QD laser will be discussed later.

The modulation bandwidth of a few hundred GHz for nanoLEDs is observed. The maximum bandwidth of a given V_{n} occurs at an certain *Q*, which can be understood by the equation [7,9]:

Since *R*_{sp,m} can be enhanced by *Q* and V_{n}, so can *τ*′_{sp,Δn}. With the increase of *Q*, *τ*_{p} increases, but *τ*′_{sp,Δn} decreases. These opposite dependences of *τ*_{p} and *τ*′_{sp,}_{Δ}* _{n}* on

*Q*are shown in Fig. 3(a).

On the other hand, the larger V_{n} results in smaller *R*_{sp,m}, and hence longer *τ*′_{sp,Δn}, which causes a smaller bandwidth of nanoLEDs. As *Q* continues to increase, *τ*_{p} limits the bandwidth. This relationship explains why bulk and QW nanoLEDs have the same bandwidth when *Q* is high. We do not express a simple formula for the optimized *Q* since *Q* is implicitly embedded in the Lorentzian function of Eq. (26) in Appendix 1.

The enhanced spontaneous emission rate also impacts the light output power versus current (L-I curve), which is shown in Fig. 4. Figure 4(a) shows that the total power of a nanoLED can be enhanced by a small volume. Furthermore, Fig. 4(b) shows the powers from the spontaneous emission and the stimulated emission. As expected, the spontaneous emission power is enhanced more by a small volume, but the stimulated emission power increases with volume since a large volume alleviates the spectral hole burning. Figure 4(c) shows the dependence of the total power on *Q* when V_{n} is 0.006. To understand this trend, we decompose the total power into the spontaneous emission power and the stimulated emission power, and they are shown in Fig. 4(d). As *Q* increases, the stimulated emission power decreases as a result of the smaller radiation loss. The spontaneous emission power increases with *Q* from 100 to 1000 while its counterparts for *Q*=1000 and 10000 overlap with each other because Γ* _{c}* is much narrower than Γ

*. Therefore, the effect of*

_{cv}*Q*on the spontaneous emission diminishes. Consequently, the total power at

*Q*= 1000 is the largest among three conditions. According to the analysis above, to design a nanoLED with high bandwidth and power,

*Q*should range from 100 to 1000, and V

_{n}should be as small as possible.

In the lasing regime, the maximum bandwidth of a laser is determined by the 3-dB maximum relaxation frequency [21, 22]:

*Q*is low, the device has to work with a high material gain, so it has a smaller differential gain

*g*′. The small

*g*′ outweighs small

*τ*

_{p}and produces a larger

*K*. Consequently, the maximum bandwidth can only reach tens of GHz. When

*Q*is further increased, the threshold material gain drops, so

*g*′ can be built up although

*τ*

_{p}increases. Figure 3(b) shows the dependences of

*τ*

_{p}and the second term in Eq. (19) on

*Q*. There exists an optimized

*Q*for different gain media. As we show, the QW laser has a greater

*f*than that of the bulk laser due to its higher differential gain. QD lasers have the maximum differential gain among three gain media, thus, the largest

_{r,max}*f*. Similar to nanoLEDs, however,

_{r,max}*τ*

_{p}limits the bandwidth when

*Q*is high.

Figure 5 exhibits the modulation responses and the corresponding spontaneous and stimulated emissions as a function of current for different *Q*’s and V_{n}’s in metal-cavity QW nanoLEDs. Figure 5(a) and 5(c) show the modulation responses of nanoLEDs with the same V_{n} but different *Q*’s. Figure 5(a) presents a larger bandwidth due to a smaller *Q* of 250 than 875 of Fig. 5(c). The bandwidths of Fig. 5(a) and 5(c) are 148 GHz and 129 GHz, respectively. We also notice that there is no relaxation resonance in the response since the stimulated emission is not dominant, which is shown in Fig. 5(b) and 5(d), respectively. In this circumstance, the carrier lifetime and the photon lifetime determine the bandwidth. The similar spontaneous emission rates in Fig. 5(b) and 5(d) are due to the saturation of *Q*, which is explained above. It also distinguishes our model from the linear model in [7, 8] and shows similar results in [9]. With the increase of current, the bandwidth *decreases* due to the increase of the damping factor.

Figures 5(e)–5(h) are the results for nanolasers with the same V_{n} of 37 but different *Q*’s. One is 875 and the other is 5000. In the laser regime, the stimulated emission dominates, as shown in Fig. 5(f) and 5(h). Under this condition, the bandwidth can be restricted by *g*′ and *τ*_{p}. Since the stimulated emission is present, the relaxation resonance appears. When
$2{\omega}_{r}^{2}$ gets closer to *γ*^{2} with the bias, the response becomes flat, which is shown in Fig. 5(e) and 5(g), respectively. Beyond that bias, *γ*^{2} will be greater than
$2{\omega}_{r}^{2}$ and the response starts to damp.

The opposite dependences of *τ*_{p} and *g*′ on *Q* offer an optimized *Q* to reach the minimum *K* and hence the *f _{r,max}* for a given condition. As a result, when

*Q*is close to 1000, the bandwidth can go to 100 GHz for MQWs, and 70 GHz for bulk material. For example, in a QW laser when V

_{n}is 0.5 and

*Q*is 875, the maximum bandwidth occurs at

*g*′ = 1.15 × 10

^{−15}cm

^{2},

*g*= 1350 cm

^{−1}, and

*S*

_{0}= 2.55 × 10

^{16}cm

^{−3}. If

*Q*= 125, the maximum bandwidth occurs at

*g*′ = 5.18 × 10

^{−16}cm

^{2},

*g*= 6280 cm

^{−1}, and

*S*

_{0}= 2.75 × 10

^{15}cm

^{−3}. Both have almost the same damping factor

*γ*since

*τ*

_{nr,Δn}and

*τ*

_{sp,Δn}are large enough to be ignored. But ${\omega}_{r}^{2}$ in the former is greater than that of the latter by three times, which enhances the bandwidth. As

*Q*continues to increase, the photon lifetime restricts the modulation bandwidth.

However, designing a *Q* for the *f _{r,max}* of a given structure requires thorough calculations since the two terms in

*K*are bias dependent. We notice that the QW lasers have a larger bandwidth than that of bulk lasers. The reason is that if a QW laser is well designed, it has a greater

*g*′ and

*S*

_{0}due to more symmetrical conduction and valence bands; both make

*K*smaller than that of bulk material lasers. This allows QW lasers to achieve a higher maximum bandwidth.

For QD cavities, due to high dot density, high confinement provided by metal cavity, and uniform dot sizes, Γ_{E} and the gain can be sustained sufficiently high. In addition, *β*_{sp}(*n*)*R*_{sp}(*n*) for QD has to be revised as [20, 27]:

*N*,

_{QD}*h*,

*g*, and

_{i}*σ*are the QD density (cm

^{−2}), the thickness of a QD layer, the degeneracy for different dot states, and the Gaussian linewidth parameter from inhomogeneous broadening, respectively;

*i*indicates the quantized states.

The metal-cavity QD light-emitting devices behave as lasers even when *Q* is small. Because of the three–dimensional confinement of electrons and holes, the high differential gain makes *K* smaller than those of bulk and QW lasers. Therefore, an unprecedented high bandwidth around 320 GHz is shown in Fig. 2(c). The effect of Γ* _{cv}* on metal-cavity QD nanolasers is also investigated. A larger Γ

*reduces the gain, and the differential gain drops accordingly. Thus, the bandwidths decrease in Fig. 2(d). From Fig. 2, we also see that the optimized*

_{cv}*Q*’s for the maximum bandwidth of bulk, QW, QDs with Γ

*=10 meV, and QDs with Γ*

_{cv}*=20 meV are around 875, 750, 250, and 375, respectively, which shift downward along*

_{cv}*Q*. The reason is that in a low

*Q*cavity, bulk and QW lasers have a large threshold material gain and most of injected carriers start to occupy at higher states (high

**K**state) and make less contribution to the transition energy of interest, which leads to a small differential gain. For QD lasers, only at an extremely small

*Q*cavity, the gain is close to the maximum, and the differential gain is reduced, thus, reducing the maximum bandwidth. It shows the importance of the multi-dimensional carrier confinement, which efficiently uses injected carriers for stimulated emission at the wavelength of interest but does not waste them in other optical transition energies. With the coincidence between the gain peak and the cavity resonance, the active medium can have a high gain and high differential gain. This explains why quantum dots are considered an excellent candidate for future light sources. In a high

*Q*condition, the photon lifetime limits the modulation response whatever gain medium is used; therefore, the bandwidths of all nanolasers decrease and converge, as expected.

#### 3.2. Scaling laws and energy per bit

Since low power consumption is an advantage of nanolasers, it is instructive to explore the scaling laws of the threshold current, I_{th}, in terms of V_{n} and *Q*. The results are shown in Fig. 6. Notice that the high threshold current of QD nanolasers result from the high surface recombination lifetime in the paper and it does not reduce as the radius increases, such as in bulk and QW cases. 0.05 ns is used for the conservative estimation to I_{th}. With the increase of V_{n} for the same *Q*, I_{th} increases. Since V* _{n}* is proportional to V

_{a}when Γ

_{E}is given. On the other hand, however, if V

_{a}is given, then different cavity designs produce different Γ

_{E}’s, thus, V

_{n}’s. Obviously, a small V

_{n}produces a small I

_{th}if

*Q*is sustained. We can see the same trend in Fig. 6. Therefore, designing a cavity with a small V

_{n}is essential for nanolasers. For the same V

_{n}, I

_{th}decreases with the increase of

*Q*due to the smaller threshold carrier density. Because of the nonlinear relation between the gain and the carrier density, I

_{th}can significantly increase as

*Q*decreases.

Using nanocavity light emitting devices to replace the electrical interconnects intensively has been investigated [1]. One important issue is the power consumption and the other is the bandwidth, and both are related to the energy per bit, which is the ratio of the electrical power,the product of voltage and current, at which the maximum bandwidth, *f _{r,max}*, occurs to the maximum bandwidth. However, nanolasers provide narrower linewidth, more coherent signal, and less phase noise than nanoLEDs [7]. Therefore, we focus on nanolasers although nanoLEDs have low energy per bit as well. The target for the energy per bit in the near future is 50 fJ/bit, but the ultimate goal is 10 fJ/bit [1]. To evaluate the potential of nanolasers used in optical interconnects, we calculate the energy per bit for the metal-cavity nanolasers using three types of gain media and the results are shown in Fig. 7. We obtain that current based on our rate equations and assume the corresponding voltage as 3 V for all devices. Since

*f*does not depend on the volume but the current does, the energy per bit monotonically increases with the volume. In addition, the energy per bit decreases with

_{r,max}*Q*for a given V

_{n}and gain medium. A low

*Q*cavity has a larger threshold current and lower bandwidth, which leads to the highest energy per bit. An optimized

*Q*cavity has the maximum bandwidth but it happens at a higher bias, which is around 15 times I

_{th}because of a smaller damping factor. Thus, the energy per bit is moderate. A high

*Q*cavity has a moderate bandwidth but the maximum bandwidth occurs at 4∼5 times I

_{th}and I

_{th}is the minimum. Therefore, a high

*Q*cavity has the minimum energy per bit. However, for QD lasers, because of three and eight quantized states in the conduction band and the valence band, respectively, the current at which the maximum bandwidth occurs for high

*Q*cavities is larger than those of bulk and QW lasers. Therefore, the energy per bit is larger. On the other, that current decreases more slowly than the maximum bandwidth, which causes the increase of the energy per bit with

*Q*.

According to Fig. 7, when V_{n} is less than 20, QD lasers with Γ* _{cv}*=10 meV have the broadest range for

*Q*to allow the energy per bit less than 50 fJ/bit. Although bulk lasers exhibit the comparable performance with others, they have smaller bandwidths and are sensitive to temperature because of no confinement of carriers. Therefore, they may not be practical in real applications. Moreover, QD lasers are less sensitive to thermal effect, which will be addressed in the next section, so we expect that QD lasers are more favorable for practical applications. However, if the dot density decreases, the advantages mentioned above diminishes. Since the small dot density reduces Γ

_{E}, leading the higher threshold gain and smaller differential gain. Therefore, I

_{th}increases and

*f*decreases. The energy per bit increases accordingly. For example, when the dot density is 10

_{r,max}^{11}cm

^{−2},

*f*can only reach 70 GHz and behave as LEDs when

_{r,max}*Q*is less than 1000. If the dot density is 10

^{10}cm

^{−2}, a higher

*Q*is needed for lasing and the bandwidth is around 30 GHz since it is restricted by

*Q*. Therefore, the high dot density is critical for QD nanolasers. Based on the assumptions above, to have the bandwidth more than 70 GHz and to meet the criterion of 50 fJ/bit, V

_{n}has to be smaller than 20 and varies with different gain media. To meet the ultimate goal of 10 fJ/bit, the normalized volume has to be less than 5, which is a challenge but may be achievable with electron-beam lithography.

## 4. The thermal effects on the modulation response

Figure 2 and 7 are carried out without the thermal effect. Although a low *Q* cavity expectedly provides a high bandwidth, a high carrier density is simultaneously needed. Thus, the thermal issue comes into play. The broadening of the Fermi distribution function due to temperature rise reduces the gain and the photon density [18]. Accordingly, the rollover in the L-I curve can occur at a low bias, while the maximum bandwidth occurs at more than ten times the threshold current, which makes the bandwidth of 100 GHz not achievable at a low *Q* cavity at room temperature. In order to alleviate the issue, there are several approaches. One way is to improve Γ_{E}; therefore, the threshold carrier density can be decreased even though *Q* is low. On the other hand, if the gain material can be insensitive to temperature, the device performance can be sustained even at high temperatures. Thus, quantum dot is a good candidate to solve the problem due to the three-dimensional confinement of carriers.

To address the insensitivity of QD nanolasers to temperature, the structure in Fig. 1(b) is used, where the radius is 0.5 *μ*m, and the numbers of DBR pairs on the top and the bottom are 5.5, respectively. The MQWs consisting of five pairs of In_{0.21}Ga_{0.79}As quantum wells and GaAs_{0.88}P_{0.12} barriers is used to compared with QD. The detail on th MQW gain modeling can be found in [18], where an excellent agreement between our theory and experiment with the thermal effect has been shown. With an extra half DBR pair adjacent to the metal, a high reflectivity can be sustained, and the volume can be effectively reduced. The FDTD method shows that *Q*, *Q*_{rad}, and Γ_{E} are 2098, 5378, and 0.026, respectively, while V_{n} is 250.

The parameters used for the modeling of the L-I curve of QW lasers are the same as those in [18]. The results are shown in Fig. 8. Figure 8(a) shows that the rollover occurs at a low bias when the thermal effect is considered. The rollover results from the detuning between the gain peak and the cavity resonance wavelength, and the broadening of Fermi distribution, which further reduces the gain, and causes the drop in power. However, if the temperature is sustained at 300 K, the rollover can be eliminated. Figures 8(b) and 8(c) show the modulation responses of two circumstances. With the thermal effect, the photon density drops at low bias, and the bandwidth is reduced accordingly. The bandwidth goes only to 28 GHz. If the temperature is constant inside the cavity, the bandwidth goes up, with the bias, to 38 GHz. This example gives the result that when the thermal effect is considered, the maximum bandwidth drops.

In contrast to QW lasers, QD lasers show the best performance, and the result is shown in Fig. 8(d). For the calculation, we set Γ* _{cv}* as 20 meV and intentionally increase the temperature from 300 K to 350 K, and do not observe the rollover in the L-I curve. The power is high due to the uniform QD size and a high QD density. The slightly larger I

_{th}results from the smaller Γ

_{E}. In addition, the bandwidth goes beyond 50 GHz even when temperature rises. Such a good performance is a benefit of the good confinement of carriers, and the large subband energy spacing between the ground state and excited states. Therefore, we favor QD nanolasers since they meet the requirements for practical applications such as low power consumption, high bandwidth, and temperature insensitivity. In addition, the energy per bit for Fig. 8(b), (c), and (d) are 569 fJ/bit, 413 fJ/bit, and 327 fJ/bit, respectively. These high values result from a low Γ

_{E}and a large active volume. To further reduce the energy per bit of the structure, a smaller diameter has to be used. In addition, more quantum well layers help to increase Γ

_{E}. For example, 20 pairs of quantum wells can be separated into 4 groups to align with the standing wave in the cavity, which has hybrid mirrors consisting of 2.5 pairs of DBR and metal. Under this condition, the laser has a Γ

_{E}around 0.187. On the other hand, quantum dot can replace quantum well and reduce the energy per bit. One significant difference from the metal nanocavity laser is that this structure is more favorable for quantum well and quantum dot gain media, since the standing wave enhancement factor further helps to increase the confinement factor, thus reducing the threshold material. With a good design, a low threshold material gain and a high differential gain are anticipated to be obtained in this metal-cavity QD surface-emitting nanolaser, which is a step toward the most energy-efficient nanolaser.

## 5. Conclusion

In conclusion, we derive the single-mode spontaneous emission rate from Fermi’s golden rule, and show that the Purcell effect is naturally included in the formulation, instead of being phenomenologically placed in the rate equations. To avoid the saturation of the spontaneous emission, a narrow linewidth of the transition energy is demanded. In addition, a small V_{n} enhances the spontaneous emission rate. In this aspect, the nanocavity is beneficial for the purpose. With rigorous derivations and definitions for each term in the rate equations, our theory is applicable to different types of cavities and dispersive materials from nano- to micro- and macroscale lasers and LEDs.

In terms of a nanolaser, the volume effect does not impact the bandwidth, but we show that there exists an optimized *Q* for a given active material. In addition, the high bandwidth of metal-cavity QD nanolasers is theoretically predicted based on submonolayer grown QDs with a high quantum-dot density *N _{QD}*, and a small full linewidth of inhomogeneous broadening parameter

*σ*. Such a high bandwidth is enhanced by a result of a high differential gain. Toward a low energy per bit, a low

*Q*, small V

_{n}, and a high Γ

_{E}are needed. The first offers a short photon lifetime, the second implies a low threshold current, and the last gives a less threshold carrier density, i.e., a larger differential gain. The metal-cavity nanolaser has the flexibility to be optimized for different gain media, and it can tailor

*Q*by adjusting the radius and insulator thickness. With the optimized design, the energy per bit less than 50 fJ/bit can be expected.

In addition, the results shown in Fig. 2, 6, and 7 vary with Γ_{E}. When Γ_{E} is reduced, the differential gain of all types of gain media reduces and the threshold current increases. Therefore, the maximum bandwidth is decreased and the energy per bit increases. Moreover, the optimized *Q* has to become larger and the bandwidth eventually is limited by the photon lifetime. If *β*_{sp}(*n*) is smaller, the energy per bit increases as well because of the increase of the threshold current.

Furthermore, we consider the thermal effects on our proposed metal-cavity VCSELs using QWs and QDs. We demonstrate that QDs are less sensitive to temperature, and the bandwidth can go beyond 50 GHz even when temperature rises during the current injection, thus demonstrating the potential advantages of using QDs in the nanolasers.

This paper addresses the direction toward a high energy-efficient laser by comparing different gain media. Other issues, such as the carrier transport phenomenon and the parasitic electrical characteristic, are also important and can limit the bandwidth while they are beyond the scope of this paper.

## Appendix 1. Derivation of single-mode spontaneous emission rate

Since single-mode spontaneous emission plays an important role in nanoLEDs, a complete understanding is important. To derive the single- mode spontaneous emission rate of the *m*-th resonant mode in a cavity, we start with Fermi’s golden rule [11, 20, 21]:

*c*indicates the conduction bands with two spins;

*v*indicates the heavy-hole bands with two spins, the light-hole bands, and two spin-orbital split-off bands;

**k**

*and*

_{c}**k**

*are the conduction and valence wave vectors, respectively;*

_{v}*μ*is the interband dipole moment;

*f*

_{c,kc}and

*f*

_{v,kv}are the occupation probabilities of electrons in the conduction band and valence band, respectively;

*E*is the transition energy; and

_{c,v}*h*̄

*ω*is the corresponding optical energy of the

_{m}*m*-th resonant mode.

The optical electric field of the *m*-th resonant mode is expressed as:

*ℰ*is the electric-field phasor of the optical mode and

_{m}**E**

*can be carried out from numerical tools such as the FDTD or FEM method.*

_{m}The normalization condition below has to be satisfied [11]:

*ε*

_{R}(

**r**,

*ω*

**) and**

_{m}*ε*

_{g}(

**r**,

*ω*

**) are the real part of the relative permittivity and relative group permittivity, defined in Eq. (4), respectively.**

_{m}Considering the finite linewidth of the transition energy in a real active material and the broadening of the modal optical mode density, *ρ*(*h*̄*ω*), both have to be replaced by a Lorentzian function, i.e.:

Since the convolution of two Lorentzian functions is still a Lorentzian function, the integral in the last term can be simplified. Furthermore, using the Fourier transform and Parseval’s theorem [11], Eq. (25) can be rewritten as:

**K**equals (

**k**

*+*

_{c}**k**

*)/2. Notice that Eq. (23) has been used in the above derivation in Eq. (26).*

_{v}## Acknowledgments

This work was sponsored by the DARPA NACHOS Program under Grant No. W911NF-07-1-0314. We thank many insightful discussions with Dr. Shu-Wei Chang, Chien-Yao Lu, and Akira Matsudaira at the University of Illinois at Urbana-Champaign. We also thank Professor Dieter Bimberg at Technical University of Berlin for encouragement in this research.

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