Large Signal Circuit Model of Two-Section Gain

advertisement
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114207
Large Signal Circuit Model of Two-Section Gain Lever Quantum Dot Laser
Ashkan Horri1** , Seyedeh Zahra Mirmoeini2 , Rahim Faez3
2
1
Young Researchers Club, Arak Branch, Islamic Azad University, Arak, Iran
Department of Electrical Engineering, Arak Branch, Islamic Azad University, Arak, Iran
3
Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
(Received 20 July 2012)
An equivalent circuit model for the design and analysis of two-section gain lever quantum dot (QD) laser is
presented. This model is based on the three level rate equations with two independent carrier populations and a
single longitudinal optical mode. By using the presented model, the effect of gain lever on QD laser performances
is investigated. The results of simulation show that the main characteristics of laser such as threshold current,
transient response, output power and modulation response are affected by differential gain ratios between the
two-sections.
PACS: 42.55.Px, 78.67.Hc
DOI: 10.1088/0256-307X/29/11/114207
Quantum dot lasers have attracted much attention
in recent years because they exhibit excellent properties such as low threshold current, high modulation bandwidth, and low frequency chirp.[1,2] In recent
studies, the gain-lever effect[3,4] is a method used to
enhance the efficiency of amplitude modulation (AM)
and optical frequency modulation (FM) at microwave
frequencies by taking advantage of the sub-linear nature of the gain versus carrier density.
In this Letter, we describe a circuit level implementation of a two-section gain lever QD laser. This
circuit model is based on the three level rate equations. The main advantages of the circuit modeling
approach include that the circuit model gives an intuitive idea of the physics of the device, and helps us to
understand the static and dynamic behavior of laser.
Indeed, circuit modeling is known as a useful approach
for design and analysis in optical systems.[5−10] The
agreement of simulation results with experimental results verifies the accuracy of our model.
The gain-lever effect can be realized in a twosection device as shown in Fig. 1.[3] Using asymmetric
current injection, the short section, which is referred
to as the modulation section, is dc-biased at a lower
gain level than the long section, termed the gain section. This scheme provides a high differential gain
under small signal rf modulation. The gain section
is only dc-biased and supplies most of the amplification but at a relatively smaller differential gain. The
fractional length of the long section is denoted by β„Ž.
Due to the gain clamping at threshold and the dependence of gain with carrier density, small changes
in carrier density in the short section produce a drastically larger variation in carrier density in the long
section to maintain the threshold gain condition. The
outcome is that the modulation efficiency and 3-dB
bandwidth depend on the differential gain ratios between the two sections.[3] Next, we present a circuit
model for the two-section QD laser in order to analyze the characteristics of the device in HSPICE. The
rate equations for semiconductor lasers are used to
investigate the static and dynamic characteristics of
lasers. The rate equations for the two-section QD laser
are[3−4]
𝐼1
𝑁1
𝑑𝑁 1
=
−
− 𝑣𝑔 1 𝑆,
(1)
𝑑𝑑
π‘ž
𝜏1
𝐼2
𝑁2
𝑑𝑁 2
=
−
− 𝑣𝑔 2 𝑆,
(2)
𝑑𝑑
π‘ž
𝜏2
𝑆
𝑑𝑆
= à 𝑣𝑔 1 (1 − β„Ž)𝑆 + à 𝑣𝑔 2 β„Žπ‘† −
𝑑𝑑
πœπ‘
(︁ 𝑁
𝑁2 )︁
1
+𝛽
+
,
(3)
𝜏1
𝜏2
where for section 𝑖, 𝑁𝑖 is the carrier number, 𝐼𝑖 is the
injection current, πœπ‘– is the carrier lifetime, 𝑔𝑖 is the material optical gain, Γ𝑖 is the optical confinement factor,
𝑆 is the photon numbers, 𝑣 is the group velocity, πœπ‘
is the photon lifetime, 𝛽 is the spontaneous emission
factor, β„Ž is the fractional length of gain section, and π‘ž
is the electron charge.
We define an approximate formula for the optical
gain as[5,6]
𝑔𝑖 = 𝐴𝑖 (𝑁𝑖 − 𝑁tri ),
(4)
where 𝐴𝑖 is the differential optical gain and 𝑁tri is
the carrier number at transparency for section 𝑖. The
inverse photon lifetime is defined as
πœπ‘−1 = 𝑣à [(1 − β„Ž)𝑔 1 + β„Žπ‘” 2 ].
(5)
We define the carrier population in the short and long
cavity sections using[5,6]
(︁ π‘žπ‘‰ )︁
1
,
(6)
𝑁1 = 𝑁e1 exp
πœ‚1 π‘˜π‘‡
(︁ π‘žπ‘‰ )︁
2
𝑁2 = 𝑁e2 exp
.
(7)
πœ‚2 π‘˜π‘‡
In this case, 𝑁e1 and 𝑁e2 are the equilibrium carrier
numbers in the short and long sections, respectively,
while πœ‚1 and πœ‚2 are the corresponding diode ideality
factor, typically set equal to 2.[5,6] 𝑉1 and 𝑉2 are the
voltages across the laser. In order to eliminate the incorrect solution regimes, we transform 𝑆 into a new
** Corresponding author. Email: [email protected]
© 2012 Chinese Physical Society and IOP Publishing Ltd
114207-1
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114207
π‘žπ‘ e1
𝐼C11 = 𝐼C12 =
,
2𝜏 1
(︁ π‘žπ‘‰ )︁
π‘žπ‘ e1 (︁
1
exp
−1
𝐼D12 =
2𝜏 1
πœ‚1 π‘˜π‘‡
(︁ π‘žπ‘‰ )︁ 𝑑𝑉 )︁
2π‘žπœ 1
1
1
+
,
exp
πœ‚1 π‘˜π‘‡
πœ‚1 π‘˜π‘‡ 𝑑𝑑
)︁
(︁ π‘žπ‘‰ )︁
π‘žπ‘ e2 (︁
2
𝐼D21 =
−1 ,
exp
2𝜏 2
πœ‚2 π‘˜π‘‡
π‘žπ‘ e2
𝐼C21 = 𝐼C22 =
,
2𝜏 2
(︁ π‘žπ‘‰ )︁
π‘žπ‘ e2 (︁
2
exp
−1
𝐼D22 =
2𝜏 2
πœ‚2 π‘˜π‘‡
)︁
(︁
2π‘žπœ 2
π‘žπ‘‰ 2 𝑑𝑉 2 )︁
+
,
exp
πœ‚2 π‘˜π‘‡
πœ‚2 π‘˜π‘‡ 𝑑𝑑
2
variable π‘‰π‘š via 𝑆 = (𝑉 π‘š + 𝛿) , where 𝛿 is an arbitrary constant. We assume that the value of 𝛿 is about
10−60 . This transformation enables the simulation to
converge to a correct numerical solution.[5,6]
i1
I2
I1
οœ±Φ“h
Gain
h
A2
g1
g2
A1
N1
N2
Carrier
Density
(19)
(20)
(21)
2
(22)
2
𝐡s2 = π‘žπ›Ό2 (Θ 2 𝐼T2 )(π‘‰π‘š + 𝛿) ,
(23)
𝑑𝑉 π‘š π‘‰π‘š
πΆπ‘β„Ž
+
= π΅π‘Ÿ1 + π΅π‘Ÿ2 + 𝐡𝑠21 + 𝐡𝑠22 ,
𝑑𝑑
π‘…π‘β„Ž
(24)
πœπ‘ 𝛽1 Θ1 𝐼𝑇 1
,
(25)
π΅π‘Ÿ1 =
𝜏1 (𝑉 π‘š + 𝛿)
πœπ‘ 𝛽2 Θ2 𝐼𝑇 2
π΅π‘Ÿ2 =
,
(26)
𝜏2 (𝑉 π‘š + 𝛿)
𝛿
𝐡𝑠21 = Γ (1 − β„Ž)𝜏 𝑝 𝛼1 (Θ 1 𝐼T1 )(π‘‰π‘š + 𝛿) − ,
2 (27)
Substituting Eqs. (6) and (7) into Eqs. (1) and (2),
and applying appropriate manipulations, we obtain
(︁ π‘žπ‘‰ )︁
(︁ π‘žπ‘‰ )︁ 𝑑𝑉 )︁
π‘žπ‘ e1 (︁
2π‘žπœ 1
1
1
1
exp
−1+
exp
2𝜏 1
πœ‚1 π‘˜π‘‡
πœ‚1 π‘˜π‘‡
πœ‚1 π‘˜π‘‡ 𝑑𝑑
(︁ π‘žπ‘‰ )︁
)︁ π‘žπ‘
π‘žπ‘ e1 (︁
1
e1
+
exp
−1 +
2𝜏 1
πœ‚1 π‘˜π‘‡
𝜏1
2
+ π‘žπ›Ό1 (𝑁 )(𝑉 π‘š + 𝛿) ,
(8)
(︁ π‘žπ‘‰ )︁ 𝑑𝑉 )︁
(︁ π‘žπ‘‰ )︁
π‘žπ‘ e2 (︁
2π‘žπœ
2
2
2
2
exp
𝐼2 =
exp
−1+
2𝜏 2
πœ‚2 π‘˜π‘‡
πœ‚2 π‘˜π‘‡
πœ‚2 π‘˜π‘‡ 𝑑𝑑
(︁ π‘žπ‘‰ )︁
)︁
π‘žπ‘ e2 (︁
2
+
exp
−1
2𝜏 2
πœ‚2 π‘˜π‘‡
π‘žπ‘ 𝑒2
2
+
+ π‘žπ›Ό2 (𝑁 )(𝑉 π‘š + 𝛿) ,
(9)
𝜏2
𝛿
𝐡𝑠22 = Γ β„Žπœ 𝑝 𝛼2 (Θ 2 𝐼T2 )(π‘‰π‘š + 𝛿) − ,
2
πΆπ‘β„Ž = 2𝜏 𝑝 , π‘…π‘β„Ž = 1.
+ I1
IC11
D11
By using the new variable π‘‰π‘š in Eq. (3), one can write
V1
𝑑𝑉 π‘š
+ π‘‰π‘š
𝑑𝑑
= Γ (1 − β„Ž)𝜏 𝑝 𝛼1 (𝑁1 )(π‘‰π‘š + 𝛿)
D12
ID11
IT1
2𝜏 𝑝
+ Γ β„Žπœ 𝑝 𝛼2 (𝑁2 )(π‘‰π‘š + 𝛿)
𝛽 1 𝑁1
𝛽2 𝑁2
+ πœπ‘
+ πœπ‘
− 𝛿,
𝜏1 (π‘‰π‘š + 𝛿)
𝜏2 (𝑉 π‘š + 𝛿)
(18)
𝐡s1 = π‘žπ›Ό1 (Θ 1 𝐼T1 )(π‘‰π‘š + 𝛿) ,
Fig. 1. Schematic view of the two-section gain-lever quantum dot laser.
𝐼1 =
(17)
(28)
(29)
IC12
ID12
Bs1
VT1
-
+ I2
D21
V2
(10)
D22
ID22
ID21
IT2
IC22
Bs2
VT2
-
where
Vm
𝛼1 (𝑁1 ) = 𝑣𝐴1 (𝑁1 − π‘π‘‘π‘Ÿ1 ), 𝛼2 (𝑁2 ) = 𝑣𝐴2 (𝑁2 − π‘π‘‘π‘Ÿ2 ).
(11)
After setting Θ1 = 2πœπ‘ž 1 , Θ2 = 2πœπ‘ž 2 and using the fact
that 𝑁1 = Θ1 𝐼T1 and 𝑁2 = Θ2 𝐼T2 , we can define
𝐼1 = 𝐼T1 + 𝐼D12 + 𝐼C12 + 𝐡s1
𝐼T1 = 𝐼D11 + 𝐼C11 ,
𝐼2 = 𝐼T2 + 𝐼D22 + 𝐼C22 + 𝐡s2 ,
𝐼T2 = 𝐼D21 + 𝐼C21 ,
(︁ π‘žπ‘‰ )︁
)︁
π‘žπ‘ e1 (︁
1
𝐼D11 =
exp
−1 ,
2𝜏 1
πœ‚1 π‘˜π‘‡
(12)
(13)
(14)
(15)
(16)
Sout
Br1
Rph
Cph
Br2
Bs21
Bs22
+
- Sout
Fig. 2. Large signal circuit model of the two-section gain
lever quantum dot laser.
The previous equations can be mapped directly
into a circuit as shown in Fig. 2. Diodes 𝐷11 and
𝐷12 , and current sources 𝐼C11 and 𝐼C12 model the
114207-2
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114207
Output photon numbers (106)
1.5
A1/A2=1
A1/A2=2
A1/A2=3
A1/A2=4
1
0.5
0
0
0.01
0.02
0.03
0.04
4
A1/A2=1
A1/A2=2
3
A1/A2=3
2
1
0
0
1
2
3
4
5
Time (10-10 s)
Fig. 5. Transient response of the two-section QD laser for
𝐴1 /𝐴2 = 1, 2, 3.
Based on the three-level rate equations, we have
proposed an equivalent circuit model of a two-section
QD laser for analysis of the steady state and dynamic
responses. By using the presented model, the effects
of differential gain ratio on laser performance, such as
threshold current, transient response, output power
and modulation response are investigated. Simulation
results show that the differential gain ratio has considerable effects on these characteristics. The results
indicate that the presented circuit model can be a very
useful tool for simulation of a two-section QD laser in
an optoelectronic device simulator.
2.5
2
creases for higher differential gain ratio 𝐴1 /𝐴2 . Figure
4 shows the frequency response of the QD laser for different values of differential gain ratio 𝐴1 /𝐴2 . It can
be found that, by increasing the differential gain ratio, the modulation bandwidth is extended. Indeed, it
has been observed theoretically that the modulation
efficiency enhancement is proportional to the ratio of
differential gains of the two sections.[8] The effect of
differential gain ratio on the transient response of QD
laser is shown in Fig. 5. It is observed that the increment in differential gain ratio leads to decrease of the
switch on delay time. Results obtained by using the
present circuit model are in good agreement with the
experimental results reported by other researchers.
Output photon numbers (106)
charge storage and carrier capture for the short section. Diodes 𝐷21 and 𝐷22 , and current sources 𝐼C21
and 𝐼C22 model the charge storage and carrier capture for the long section. 𝐡s1 and 𝐡s2 model the
effect of stimulated emission on the short and long
sections, respectively. 𝑅ph and 𝐢ph model the time
variation of the photon number under the effect of
spontaneous and stimulated emission, which are denoted by π΅π‘Ÿ1 , π΅π‘Ÿ2 , 𝐡s21 , and 𝐡s22 . 𝑆out produces the
output photon number of the laser in the form of a
voltage.
The device investigated was grown by MBE on
an n+ GaAs substrate. The active region consists of 10-stacked layers of InAs QDs covered by 5nm In0.15 Ga0.75 As QWs, separated by 33-nm GaAs
spacers, of which 10-nm is carbon p-type doped.
The cladding layers are step-doped 1.5-µm-thick
Al0.35 Ga0.65 As. The laser structure is capped with
a 400-nm-thick GaAs. Two-section lasers with 1.5mm cleaved cavity lengths 𝐿cav , and 3-µm-wide ridge
waveguides were fabricated. The electrical isolation
between the 0.5 and 1-mm segments was achieved using proton implantation.[3,4] The fraction length of
gain section β„Ž is 0.8. The carrier recombination lifetimes 𝜏1 and 𝜏2 are 5 ns. The spontaneous emission
factor 𝛽 is 10−5 . The circuit model of Fig. 2 is used
to analyze the characteristics of the two-section QD
laser in HSPICE.
0.05
I1 (A)
Modulaion response (dB)
Fig. 3. Output photon number of the two-section QD
laser versus differential gain ratios 𝐴1 /𝐴2 = 1, 2, 3, 4.
References
15
A1/A2=1
10
5
A1/A2=2
A1/A2=3
A1/A2=4
0
−5
−10
107
108
109
1010
Frequency (Hz)
Fig. 4. Modulation response of the two-section QD laser
for different differential gain ratios 𝐴1 /𝐴2 = 1, 2, 3, 4.
Figure 3 shows the light-current characteristics
when the differential gain ratio changes from 1 to
4. As shown in the figure, the threshold current de-
[1] Coldren L A and Corizne S W 1995 Diode Lasers Photonic
Integrated Circuits (New York: John Wiley & Sons) chap 4
p 129
[2] Sugawara M 1999 Self-assembled InGaAs/GaAs Quantum
Dots (Tokyo: Academic) chap 6 p 241
[3] Li Y, Naderi N A, Kovanis V and Lester L F 2010 IEEE
Photon. J. 2 321
[4] Moore N and Lau K Y 1989 Appl. Phys. Lett. 55 936
[5] Mena P V, Kang S M and Detemple T A 1997 IEEE J.
Lightwave. Technol. 15 717
[6] Yavari M H, Ahmadi V 2009 IEEE J. Sel. Top. Quantum
Electron. 15 774
[7] Horri A, Faez R and Hoseini H R 2011 IEICE Electron.
Express 8 245
[8] Yousefvand H R, Ahmadi V and Saghafi K 2010 IEEE J.
Lightwave Technol.
[9] Mena P V 1998 PhD Dissertation (Chicago: University of
Illinois)
[10] Horri A and Faez R 2011 Opt. Eng. 50 034202
114207-3
Download