Theory for voltage modulation of transistor lasers using Franz-Keldysh absorption in the presence of optoelectronic feedback

Chi-Hsiang Chang; Shu-Wei Chang; Chao-Hsin Wu

doi:10.1364/OE.24.025515

1. Introduction

As developments of technology and information circulation grow rapidly, optoelectronic integrated circuits (OEICs) become more important [1], and therefore the more attention is drawn to the high speed optoelectronic devices nowadays. The light-emitting transistor (LET) based on heterostructures of the indium gallium phosphide/gallium arsenide (InGaP/GaAs) material was first proposed by Feng and Holonyak in 2004 [2,3]. With both the high-speed features of heterojunction bipolar transistors (HBTs) and efficient radiative recombination of carriers in quantum wells (QWs), a modulation bandwidth of LETs as high as 4.3 GHz has been demonstrated in the regime of spontaneous emissions [4]. In addition to InGaP/GaAs, different material systems such as the indium-phosphide (InP) family which is utilized in the telecommunication were proposed and fabricated [5–7]. These unique three-port devices have great potential for future optical interconnects and OEICs.

The optical response of LETs is faster than that of conventional diode lasers (DLs) [8,9] due to a heavily doped base and transistor structure [10]. Furthermore, after the redesign of layouts and introduction of cavities for photon storage, LETs can be turned into transistor lasers (TLs) [11]. It has been shown previously that TLs could exhibit a high modulation bandwidth capable of data transmissions at 40 Gb/s [12]. As TLs are operated in the forward-active mode, the junction between the emitter and base (EB junction) is forward biased while that between the base and collector (BC junction) is reversely biased. The significant electric field in the BC junction induces additional absorption for the photons generated in the active region even though the photon energy is less than the bandgap energy of the reversely biased junction. This phenomenon is known as the Franz-Keldysh (F-K) absorption or photon-assisted tunneling. With the F-K effect in BC junctions [13], the voltage-controlled modulation of TLs becomes plausible in addition to the current-controlled modulation of typical DLs. The optical responses due to the voltage modulation are relatively flat in the experiment [14,15], which is beneficial from a system point of view. In this way, external voltage-controlled modulators which are designed for conventional devices may not be required for TLs [16,17].

In this work, the effect of F-K absorption on the voltage modulation of TLs is investigated. Due to the F-K absorption in the BC junction, the mechanism of breakdowns is enhanced in TLs [18]. To understand the dynamics and optical responses of TLs, we incorporate the F-K effect into conventional rate equations which describe the modulation characteristics of DLs. Based on the models of rate equations and F-K absorption [19,20], new terms related to the F-K effect are introduced into the rate equations. The theoretical results of direct-current (DC) and small-signal alternating-current (AC) characteristics of optical responses are both investigated. While the DC characteristics look physical, the intrinsic optical AC response of TLs under the F-K voltage modulation exhibits an enhancement peak of about 25 dB, which is scarcely observed in pervious experiment. In fact, apart from the intrinsic optical response, an electrical model that incorporates parasitic resistances, capacitances, and inductances should be considered in the overall small-signal analysis. The measured optical response does depend on these parasitic elements. These analytical techniques are often adopted for light-emitting diodes (LEDs) or LETs [21,22]. In our case of TLs, a theoretical model composed of the intrinsic optical response and an electrical transfer function which is fed back by this optical response is proposed to elucidate the true behavior of voltage modulation in TLs. The giant AC peak disappears at measurement probes through this optoelectronic feedback. With the rate equations in which the FK effect is built and the electrical transfer function taken into account, the overall voltage-controlled optical modulation response of TLs is explained.

This paper is organized as follows. The device structure and results of the DC model are presented in section 2. The small-signal analysis and results are demonstrated in section 3. In section 4, we present the complete model which describes the characteristics of voltage-controlled modulations for TLs. The conclusion is given in section 5.

2. Device structure and rate equations with Franz-Keldysh absorption

The layer structure of the device considered in this study is shown schematically in Fig. 1(a). It consists of a 3000 Å n-type doped GaAs buffer layer, followed by a 634 Å n-type Al_0.4Ga_0.6As layer, a 5000 Å n-type Al_0.9Ga_0.1As layer, and a 150 Å n-type Al_0.4Ga_0.6As layer, which form the bottom cladding layer as a whole (5784 Å in total). These layers are then followed by a 200 Å n-type sub-collector layer which is heavily doped for collector contact, a 120 Å In_0.49Ga_0.51P etch stop layer, a 600 Å undoped GaAs collector layer, and a 980 Å p-type GaAs base layer which contains a 160 Å InGaAs QW as the active region. The epitaxial structure is completed with the growth of a 250 Å n-type In_0.49Ga_0.51P emitter layer, and a 5100 Å n-type Al_xGa₁₋_xAs (x = 0.35, 0.8, and 0.98) layer for the current-confinement aperture. Finally, a 1000 Å n-type GaAs layer which is heavily doped for emitter contact is deposited on the top. The emission range of the QW is around 980 nm.

Fig. 1 The layer structure and band diagram of the TL. A QW is incorporated in the p-type base for photon emissions. The additional holes generated by the F-K absorption flow back into the base (active) region and participate in the carrier-photon interaction.

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As illustrated in Fig. 1(b), the 600 Å intrinsic GaAs layer between the base and collector function as a heterojunction phototransistor (HPT) during the voltage-controlled modulation. In the presence of the F-K absorption due to the reverse bias of the BC junction, the photons emitted from the QW are absorbed and generate additional electron-hole pairs in the region of intrinsic GaAs. These electron-hole pairs are then split by the electric field with electrons and holes flowing into the collector and base (active) region, respectively. The holes injected into the base may then participate in the carrier-photon interaction again.

In the presence of F-K absorption in the BC junction, the conventional rate equations of carrier and photon densities in DLs need to be modified. The generalized rate equations are

\frac{d N}{d t} = \frac{η_{i} i_{B}}{e V_{a}} - R - v_{g} g N_{p} + η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} α N_{p},

\frac{d N_{p}}{d t} = (Γ_{a} v_{g} g - \frac{1}{τ_{p}}) N_{p} + Γ_{a} β_{sp} R_{sp} - Γ_{J_{BC}} v_{g} α N_{p},

where

N

and

N_{p}

are the minority carrier (electron) and photon densities, respectively;

η_{i}

is the injection efficiency;

i_{B}

is the base current;

e

is the electron charge;

V_{a}

is the volume of active region;

R

is the carrier recombination rate and is modeled as the spontaneousemission rate

R_{sp} = N / τ_{sp}

parameterized by a lifetime

τ_{sp}

which depends on the p-type doping level in the base (non-radiative recombination is not considered here);

v_{g}

is the group velocity;

g = a_{0} (N - N_{tr}) / (1 + ε N_{p})

is the optical gain for which the linear gain model parameterized by the differential gain

a_{0}

, transparency carrier density

N_{tr}

, and gain compression factor

ε

is adopted;

η_{BC}

is the injection efficiency of the hole current from the collector to base;

Γ_{a}

and

Γ_{J_{BC}}

are the confinement factors of the active region and BC junction, respectively;

α

is the absorption coefficient due to the F-K effect;

τ_{p}

is the photon lifetime;

β_{sp}

is the spontaneous emission coupling factor defined as the percentage of the total spontaneous emissions coupled into the lasing mode. In these two rate equations, we adopt the formula of F-K absorption coefficient

α

in [20], which has the following analytical form:

α \propto \sqrt{ℏ θ_{F}} {- η {Ai}^{2} (η) + {[{Ai}^{'} (η)]}^{2}},

η = \frac{E_{g} - ℏ ω_{p}}{ℏ θ_{F}}, θ_{F} = {(\frac{e F}{\sqrt{2 μ ℏ}})}^{2 / 3},

where

Ai (η)

and

{Ai}^{'} (η)

are Airy function of the first kind and its derivative, respectively;

E_{g}

(1.421 eV) and

ℏ ω_{p}

(1.265 eV) are energies corresponding to the bandgap of intrinsic GaAs and lasing mode, respectively;

μ

is the reduced mass of electron-hole pairs;

F

is the electric field between the base and collector; and

ℏ

is Planck’s constant. In the DC bias condition, the electric field

F

is approximately the ratio between the voltage drop

V_{CB}

between the C and B terminals and thickness

d

of intrinsic GaAs in the BC junction, namely,

F = V_{CB} / d

. We note that the last term

Γ_{J_{BC}} v_{g} α N_{p}

in Eq. (1b) is the modal absorption rate due to the F-K effect in the BC junction. A similar term

η_{BC} (Γ_{J_{BC}} / Γ_{a}) v_{g} α N_{p}

in Eq. (1a) corresponds to the additional electrons (minority carriers) from the emitter side which balance the holes generated by the F-K absorption and injected into the active region. Numerical values of the parameters utilized in later calculations are listed in Table 1. The two confinement factors

Γ_{a}

and

Γ_{J_{BC}}

are estimated from the layer structure in Fig. 1 with commercial mode solver of waveguides. Others related to the gain medium (

a_{0}

and

N_{tr}

) areadjusted from those of the similar material system in [19], taking into account that the preexisting holes are present in the QW due to the heavily p-doped base. A lifetime

τ_{sp}

in the sub-nanosecond rather nanosecond range is adopted for TL here since the threshold current density of TLs is usually a few times higher than that of commercial DLs. This choice of

τ_{sp}

is also consistent with the higher modulation bandwidth of TLs than that of DLs [12].

Table 1. List of parameters for TLs.

View Table

The TL is operated in the common-emitter configuration. The voltage drop $V_{CB}$ changes the absorption coefficient $α$ . The DC light-versus-current-voltage (LIV) and carrier-density-versus-current-voltage family curves are self-consistently solved from Eqs. (1a) and (1b) by setting the time derivatives of $N$ and $N_{p}$ to zero (steady state) at a given $i_{B}$ and $V_{CB}$ . Figure 2(a) shows the LIV family curves of the TL. The threshold current increases with $V_{CB}$ since the larger reverse bias results in the more significant F-K absorption and hence the fewer photons and output power in the lasing mode. Meanwhile, the collector current $i_{C}$ at a high reverse bias of $V_{CB}$ would also increase as a result of the additional electrons generated from the F-K absorption. The simulated LIV curves show great consistency with the experimental results presented by Feng et al [25]. The F-K absorption and corresponding photon-assisted tunneling feedback current will play an important role in determinations of DC performance as well as AC frequency response of TLs. The carrier density $N$ in the active region as a function of $i_{B}$ and $V_{CB}$ is shown in Fig. 2(b). For DLs, the carrier density is pinned at a certain value once the threshold condition is reached. This phenomenon is also present in TLs. As the voltage $V_{CB}$ increases, this threshold carrier density is raised, in accordance with the power drop on the LIV family curves. At a fixed $i_{B}$ , as the voltage $V_{CB}$ exceeds certain critical value, the device would not lase anymore and just turns into a spontaneous-emission source.

Fig. 2 (a) The LIV family curves of the TL. The more significant F-K absorption at the larger $V_{CB}$ makes the light output turns smaller. (b) The carrier density versus $i_{B}$ and $V_{CB}$ . _.At a fixed $i_{B}$ , if $V_{C B}$ is larger than a critical value, the device would not lase anymore and turn into a spontaneous-emission source.

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3. Small-signal analysis

After the DC bias condition is determined, we then look into the small-signal analysis of the voltage-controlled modulation. For a generic variable $X$ , we write it as $X = X^{(0)} + Δ X$ , where $X^{(0)}$ and $Δ X$ are the DC component and small deviation of $X$ . In order to monitor the dynamics in the response of TLs, we carry out the expansion of various variables in Eqs. (1a) and (1b) around their biased points but only keep the perturbation up to the first order. In this way, Eqs. (1a) and (1b) are transformed into

\frac{d Δ N}{d t} = - Δ R_{sp} - v_{g} Δ g N_{p}^{(0)} + v_{g} g^{(0)} Δ N_{p} + η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} (v_{g} Δ α N_{p}^{(0)} + v_{g} α^{(0)} Δ N_{p}),

\frac{d Δ N_{p}}{d t} = Γ_{a} v_{g} Δ g N_{p}^{(0)} + Γ_{a} v_{g} g^{(0)} Δ N_{p} - \frac{Δ N_{p}}{τ_{p}} - Γ_{J_{BC}} v_{g} Δ α N_{p}^{(0)} - Γ_{J_{BC}} v_{g} α^{(0)} Δ N_{p} + Γ_{a} Δ (β_{sp} R_{sp}),

where

Δ i_{B}

is set to zero in absence of the current-controlled modulation.

In TLs, the p-type base is heavily doped, which brings about preexisting holes in the QW. Under such circumstances, the density of holes in the QW and therefore the lifetime $τ_{sp}$ does not vary much during small-signal modulations. In this way, we may approximate $Δ R_{sp}$ as $Δ N / τ_{sp}$ . Typically, the last term on the right-hand side of Eq. (3b) is small compared with others in conventional DLs, and we would simply drop it. The gain variation $Δ g$ can be expanded by both carrier and photon densities as

Δ g = a Δ N - a_{p} Δ N_{p},

a = {\frac{\partial g}{\partial N} |}_{N = N^{(0)}, N_{p} = N_{p}^{(0)}} = \frac{a_{0}}{1 + ε N_{p}^{(0)}},

a_{p} = {- \frac{\partial g}{\partial N_{p}} |}_{N = N^{(0)}, N_{p} = N_{p}^{(0)}} = \frac{ε g^{(0)}}{1 + ε N_{p}^{(0)}} .

Also, the absorption variation

Δ α

is proportional to

Δ F

through a constant

K

defined as

Δ α = K Δ F, K = \frac{α^{(0)}}{3 F^{(0)}} {\frac{η^{(0)} {Ai}^{2} (η^{(0)}) + {[{Ai}^{'} (η^{(0)})]}^{2}}{- η^{(0)} {Ai}^{2} (η^{(0)}) + {[{Ai}^{'} (η^{(0)})]}^{2}}} .

We then substitute $Δ g$ in Eq. (4a) into Eqs. (3a) and (3b) and rewrite the two rate equations in the matrix form as

\frac{d}{d t} (\begin{matrix} Δ N \\ Δ N_{p} \end{matrix}) = (\begin{matrix} - γ_{NN} & - γ_{NP} \\ γ_{PN} & - γ_{PP} \end{matrix}) (\begin{matrix} Δ N \\ Δ N_{p} \end{matrix}) + (\begin{matrix} γ_{N α} Δ α \\ - γ_{P α} Δ α \end{matrix}),

where various matrix elements (coupling factors) are

γ_{NN} \equiv \frac{1}{τ_{sp}} + v_{g} a N_{p}^{(0)}, γ_{NP} \equiv v_{g} g^{(0)} - v_{g} a_{p} N_{p}^{(0)} - η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} α^{(0)}, γ_{N α} \equiv η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} N_{p}^{(0)},

γ_{PN} \equiv v_{g} a Γ_{a} N_{p}^{(0)}, γ_{PP} \equiv v_{g} a_{p} Γ_{a} N_{p}^{(0)} - Γ_{a} v_{g} g^{(0)} + \frac{1}{τ_{p}} + Γ_{J_{BC}} v_{g} α^{(0)}, γ_{P α} \equiv Γ_{J_{BC}} v_{g} N_{p}^{(0)} .

With a sinusoidal modulation of voltage

Δ V_{CB}

, various responses are also sinusoidal. In this case, we may write the solution of a generic variation

Δ X

as

Δ X = Re [X_{1} exp (j ω t)],

where

X_{1}

is the amplitude of the modulation; and

ω

is the modulation frequency. In the sinusoidal steady state, the time derivative

d / d t

gives rise to the factor

j ω

for various amplitudes

X_{1}

, and Eq. (6) can be rearranged as

(\begin{matrix} γ_{NN} + j ω & γ_{NP} \\ - γ_{PN} & γ_{PP} + j ω \end{matrix}) (\begin{matrix} N_{1} \\ N_{p, 1} \end{matrix}) = (\begin{matrix} γ_{N α} α_{1} \\ - γ_{P α} α_{1} \end{matrix}) .

The small-signal carrier and photon densities then can be obtained by inverting the matrix at the left-hand side of Eq. (8). Their expressions are shown as follows:

N_{1} = \frac{H (ω) [(γ_{PP} + j ω) γ_{N α} + γ_{NP} γ_{P α}]}{ω_{R}^{2}} α_{1},

N_{p, 1} = \frac{H (ω) [- (γ_{NN} + j ω) γ_{P α} + γ_{N α} γ_{PN}]}{ω_{R}^{2}} α_{1},

where the transfer function

H (ω)

is featured by the relaxation resonance frequency

ω_{R}

ad damping factor

γ

as

H (ω) \equiv \frac{ω_{R}^{2}}{Δ} = \frac{ω_{R}^{2}}{ω_{R}^{2} - ω^{2} + j ω γ}, Δ = γ_{NN} γ_{PP} + γ_{NP} γ_{PN} + j ω (γ_{NN} + γ_{PP}) - ω^{2},

\begin{array}{l} ω_{R}^{2} = γ_{NP} γ_{PN} + γ_{NN} γ_{PP} \\ = (v_{g} g^{(0)} - v_{g} a_{p} N_{p}^{(0)} - η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} α^{(0)}) v_{g} a Γ_{a} N_{p}^{(0)} \end{array}

+ (\frac{1}{τ_{sp}} + v_{g} a N_{p}^{(0)}) (v_{g} a_{p} Γ_{a} N_{p}^{(0)} - Γ_{a} v_{g} g^{(0)} + \frac{1}{τ_{p}} + Γ_{J_{BC}} v_{g} α^{(0)}),

γ = γ_{NN} + γ_{PP} = \frac{1}{τ_{sp}} + v_{g} a N_{p}^{(0)} + v_{g} a_{p} Γ_{a} N_{p}^{(0)} - Γ_{a} v_{g} g^{(0)} + \frac{1}{τ_{p}} + Γ_{J_{BC}} v_{g} α^{(0)} .

The relaxation resonance frequency

ω_{R}

and damping factor

γ

in Eqs. (10b) and (10c) are different from those of conventional DLs [19] since they include the new terms from the F-K effect in the BC junction. By utilizing the expression of

Δ α

in Eq. (5) for

α_{1}

and defining an effective voltage amplitude

V_{1} \equiv F_{1} d

which is close to

V_{CB, 1}

at low frequencies, the ratio between the variation

N_{p, 1}

of photon density and effective voltage

V_{1}

, which is defined as

T_{p} (ω)

, can be derived as

T_{p} (ω) \equiv \frac{N_{p, 1}}{V_{1}} = \frac{[v_{g} a Γ_{a} N_{p}^{(0)} η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} N_{p}^{(0)} {(j ω + \frac{1}{τ_{sp}} + v_{g} a N_{p}^{(0)})}^{- 1} - Γ_{J_{BC}} v_{g} N_{p}^{(0)}] K}{[\begin{array}{l} j ω - v_{g} a Γ_{a} N_{p}^{(0)} (v_{g} a_{p} N_{p}^{(0)} - v_{g} g^{(0)} + η_{BC} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} α^{(0)}) {(j ω + v_{g} a N_{p}^{(0)} + \frac{1}{τ_{sp}})}^{- 1} \\ + v_{g} a_{p} Γ_{a} N_{p}^{(0)} - Γ_{a} v_{g} g^{(0)} + \frac{1}{τ_{p}} + Γ_{J_{BC}} v_{g} α^{(0)} \end{array}]} .

After properly normalized with the DC response (

ω = 0

), the magnitude

U (ω)

of the intrinsic optical response

T_{p} (ω)

in the logarithmic scale is defined as

U (ω) = 10 \log {| \frac{T_{p} (ω)}{T_{p} (0)} |}^{2} .

The responses $U (ω)$ in different bias conditions are shown in Fig. 3. In Fig. 3(a), we show the responses under different levels of current injections $i_{B}^{(0)}$ from 60 to 90 mA at $V_{CB}^{(0)} = 1.6 V$ . From Eq. (10b), the relaxation frequency $ω_{R}$ increases positively with the DC photon density $N_{p}^{(0)}$ . Since the larger base current $i_{B}^{(0)}$ brings about the higher photon density $N_{p}^{(0)}$ , the larger $ω_{R}$ and therefore the higher modulation bandwidth can be obtained, as can be observed in Fig. 3(a). This phenomenon is similar to the current-controlled modulation of DLs, in which the output power (photon density), relaxation resonance frequency, and modulation bandwidth would first increase with the injection current once the threshold is reached. On the other hand, under the high current operation, the photon density in factdecreases due to thermal effect for typical DLs. This effect reduces the modulation bandwidth at a high injection level. For TLs, an analogous situation could be also potentially present in the response of voltage-controlled modulation at a high injection current. This phenomenon, however, is not included in our rate-equation model because the power rollover cannot be properly addressed by the linear gain with gain compression. The responses $U (ω)$ at different $V_{CB}^{(0)}$ from 1.2 to 2.4 V but a fixed $i_{B}^{(0)}$ of 70 mA are shown in Fig. 3(b). As a result of the larger reverse bias voltage $V_{CB}^{(0)}$ , the F-K absorption becomes more significant, and the number of photons becomes smaller. Hence, the modulation bandwidth decreases as $V_{CB}^{(0)}$ increases.

Fig. 3 (a) The magnitude of the intrinsic optical response in the logarithmic scale under different $i_{B}^{(0)}$ from 60 to 90 mA at $V_{CB}^{(0)}$ = 1.6 V. The modulation bandwidth increases with $i_{B}^{(0)}$ since the photon density is raised in the cavity. (b) The counterparts at different $V_{CB}^{(0)}$ from 1.2 to 2.4 V ( $i_{B}^{(0)}$ fixed at 70 mA). The modulation bandwidth decreases with $V_{CB}^{(0)}$ due to the less photon density in the cavity.

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In various bias conditions of TLs considered above, the magnitude $U (ω)$ of the intrinsic optical response in the logarithmic scale shows an AC enhancement peak more than 20 dB around $ω = ω_{R}$ . The enhancement peak can be mainly attributed to the frequency dependency of optical modulation amplitude shown in Eq. (9b), which is similar to the frequency response of carrier densities or frequency modulation in the current modulation of conventional DLs [19]. In the case of TLs, the F-K absorption can directly modulate the photon density and bring a frequency zero ( $j ω$ ) to the numerator of Eq. (9b), which then leads to the large AC peak. This very prominent peak in the modulation response, however, was seldom observed experimentally. Next, a theoretical model composed of the intrinsic optical response and an electrical transfer function fed back by this optical response is proposed to explain the correct behavior of voltage modulation in TLs.

4. Small-signal electrical model

The modulation bandwidth of optical devices is limited by the parasitic effect of electrical circuits due to the geometry of devices. Generalizing the small-signal circuit of the hybrid $π$ model for HBTs, we illustrate the counterpart of TLs for voltage-controlled modulation in Fig. 4. In this model, $R_{π}$ and $C_{π}$ are the input resistance and capacitance as looking into the base, respectively; $R_{μ}$ and $C_{μ}$ are those due to the reversely-biased BC junction; and $R_{0}$ is a resistance due to the Early effect. An additional current source $i_{add}$ resulted from the F-K absorption in the BC junction is incorporated in this electrical model. The parameters $R_{E}$ and $L_{E}$ stand for the parasitic resistance and inductance of the emitter port while $R_{C}$ and $L_{C}$ are the counterparts of the collector port. Due to the common emitter configuration in this study, the emitter is connected to the ground. Because the base current is fixed, the base port can be regarded as open-circuited in the small-signal model. In our electrical circuit model, an AC voltage source with a magnitude of 0.01 V is connected to the external collector port (C) for small-signal modulations. The other parasitic elements are set to typical values of the small-signal model for LETs and TLs [23,24].

Fig. 4 The small-signal circuit of the TL. An additional current source which is resulted from the F-K absorption in the BC junction is incorporated in this electrical model. The admittance corresponding to this additional current can be deduced from rate equations.

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The intrinsic optical response $T_{p} (ω)$ obtained in previous calculations is in fact the modulation response related to the voltage amplitude $V_{C^{'} B,1}$ of the internal collector port ( $C^{'}$ ), namely, $V_{C^{'} B,1} = V_{1}$ . In practice, what we monitor is the modulation response corresponding to the external voltage $V_{CB,1}$ rather than that of $V_{C^{'} B,1}$ . Therefore, the overall response $T_{overall} (ω)$ which is experimentally observed should be represented as follows:

T_{overall} (ω) \equiv \frac{N_{p, 1}}{V_{CB, 1}} = \frac{N_{p, 1}}{V_{C^{'} B, 1}} \times \frac{V_{C^{'} B, 1}}{V_{CB, 1}} = T_{p} (ω) \times T_{e} (ω),

where

T_{e} (ω) \equiv V_{C^{'} B, 1} / V_{CB, 1}

is defined as the electrical transfer function which depends on various small-signal resistances as well as capacitances and would be fed back by

T_{p} (ω)

. The additional current

i_{add}

can be inferred from the rate equation of carrier density in Eq. (1a) and is expressed as

i_{add} = e V_{a} η^{'} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} α N_{p},

where

η^{'}

is the percentage of carriers surviving from the electron-hole recombination after being generated in the reversely biased BC junction. The small-signal admittance

Y (ω)

corresponding to this additional current is then deduced from Eq. (13). Through the small-signal analysis at the sinusoidal steady state, we obtain the relation between the amplitudes

i_{add,1}

and

V_{C^{'} B,1}

of the small signals

Δ i_{add}

and

Δ V_{C^{'} B}

:

i_{add,1} = Y (ω) V_{C^{'} B,1},

\begin{array}{l} Y (ω) = e V_{a} η^{'} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} {(\frac{\partial α}{\partial V_{C^{'} B}} N_{p}^{(0)} + α^{(0)} \frac{N_{p,1}}{V_{C^{'} B,1}})}_{V_{C^{'} B} \approx V_{C^{'} B}^{(0)}, i_{B} = i_{B}^{(0)}} \\ = e V_{a} η^{'} \frac{Γ_{J_{BC}}}{Γ_{a}} v_{g} [K N_{p}^{(0)} + α^{(0)} T_{p} (ω)] . \end{array}

Equation (14b) indicates that the admittance

Y (ω)

depends on the intrinsic optical response

T_{p} (ω)

. Through this admittance,

T_{p} (ω)

directly affects the electrical response

T_{e} (ω)

and therefore the overall response

T_{overall} (ω)

. In Eq. (14b), if the term proportional to response

T_{p} (ω)

dominates, the magnitude of

Y (ω)

may follow the dramatic variation of

T_{p} (ω)

shown in Figs. 3(a) and 3(b). Therefore, the magnitude of admittance

Y (ω)

would be at its maximum as the magnitude of the intrinsic optical response

T_{p} (ω)

is at its peak, which reduces the magnitude of the internal voltage drop

V_{C^{'} B,1}

across the BC junction. In other words, the magnitude of the electrical transfer function

T_{e} (ω)

is approximately at its minimum when the intrinsic optical response is at its maximal magnitude if the effects from various resistances and capacitances are relatively minor. In this model of the TL, behavior of the electrical transfer function is therefore subject to the intrinsic optical response. After taking the full circuit model into account, various normalized magnitude responses of voltage modulations for the TL in the logarithmic scale at different bias voltages (

V_{CB}^{(0)}

= 1.8, 2.0 and 2.2 V) and an injection current

i_{B}^{(0)}

of 70 mA are shown in Fig. 5. The resistance of collector is set to 3 ohm. The electrical transfer function

T_{e} (ω)

is in fact sensitive to this resistance, and this issue will be addressed next. As the magnitude of the admittance is maximal, the magnitude of the voltage drop across the admittance would be minimal. The smallest magnitude of the electrical transfer function

T_{e} (ω)

is reached under such circumstances. After the intrinsic optical response and small-signal circuit model are taken into account simultaneously, the relatively flat magnitude response of

T_{overall} (ω)

as compared to

T_{p} (ω)

can be observed. On the other hand, the electrical transfer function

T_{e} (ω)

decreases as the reversely biased voltage

V_{CB}^{(0)}

increases, as can be told from Fig. 5.

Fig. 5 The comparison among various responses related to the voltage-controlled modulations of the TL at a $V_{CB}^{(0)}$ of (a) 1.8 V, (b) 2 V, and (c) 2.2 V. The magnitude of the electrical transfer function is minimal as the magnitude of the intrinsic optical response is at its peak.

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The modulation bandwidth of voltage modulation is limited by the parasitic effect of electrical circuits, especially by the collector resistance $R_{C}$ . The normalized magnitude of the electrical transfer functions $T_{e} (ω)$ in the logarithmic scale at different collector resistances from 3 to 6 ohm are shown in Fig. 6(a), and the corresponding overall optical responses are shown in Fig. 6(b). The bias point is set at $V_{CB}^{(0)} = 2.0 V$ and $i_{B}^{(0)} = 70 mA$ . From Fig. 6(a), as the collector resistance increases, the magnitude of the voltage drop in the BC junction decreases, and therefore the magnitude to electrical transfer function $T_{e} (ω)$ is significantly reduced. As a result, the overall optical response $T_{overall} (ω)$ of the TL is sensitive to collector resistance which is related to geometry and material of device, as can be told from the smaller modulation bandwidth at the larger $R_{C}$ shown in Fig. 6(b). Therefore, there would be a trade-off between the modulation bandwidth and flat response due to the collector resistance.

Fig. 6 (a) The electrical transfer function $T_{e} (ω)$ and (b) overall response $T_{overall} (ω)$ at different collector resistances $R_{C}$ . In the TL, the response $T_{overall} (ω)$ is sensitive to the collector resistance. As the collector resistance increases, its normalized magnitude decreases but becomes relatively flat.

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5. Conclusion

Compared with conventional DLs, the TL can additionally provide the voltage-controlled modulation due to effect of F-K absorption. On the other hand, the theoretical model merely based on photon-carrier rate equations which incorporate various F-K effects leads to a huge AC enhancement peak on the voltage-controlled modulation response. This phenomenon is not consistent with the relatively flat response observed in the previous experiment. A theoretical model including not only the intrinsic optical characteristics but also an electrical transfer function fed back by optical responses is necessary to explain the correct behaviors of voltage-controlled modulation assisted by the F-K absorption.

Funding

Ministry of Science and Technology of Taiwan (MOST of Taiwan) (MOST 102-2221-E-002-192-MY3, MOST 104-2622-E-002-024-CC3, MOST 104-2622-E-002-032-CC2, MOST 104-2218-E-005-004).

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	$η_{i}$	$V_{a}$	$τ_{sp}$	$a_{0}$
Value	$0.8$	$16.08 (μ m^{3})$	$200 (ps)$	$6 .1 \times 10^{- 20} (m^{2})$
	$η_{BC}$	$Γ_{J_{BC}}$	$Γ_{a}$	$v_{g}$
Value	$0.8$	$0.3659$	$0.074481$	$7.5 \times 10^{7} (m s^{- 1})$
	$N_{tr}$	$ε$	$τ_{p}$	$β_{sp}$
Value	$7 \times 10^{23} (m^{- 3})$	$1.5 \times 10^{- 23} (m^{3})$	$2.15 (ps)$	$8.69 \times 10^{- 5}$

Theory for voltage modulation of transistor lasers using Franz-Keldysh absorption in the presence of optoelectronic feedback

Abstract

1. Introduction

2. Device structure and rate equations with Franz-Keldysh absorption

3. Small-signal analysis

4. Small-signal electrical model

5. Conclusion

Funding

References and links

Cited By

Figures (6)

Tables (1)

Equations (26)

Optics Express