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EE 232 Lightwave Devices
Lecture 15: Strained Quantum Well Laser
Reading: Chuang, Sec. 10.4
(There is also a good discussion in Coldren, Appendix 11)
Instructor: Ming C. Wu
University of California, Berkeley
Electrical Engineering and Computer Sciences Dept
Dept.
©2008. University of California
EE232 Lecture 15-1
Reduction of Lasing Threshold Current Density
by Lowering Valence Band Effective Mass
Bernard-Duraffourg Condition:
FC − FV ≥ =ω ≥ Ee1 − Eh1
Ordinary Semiconductor
Ideal Semiconductor
mh* ≈ 6me*
mh* ≈ me*
High transparenc
transparency
Lo transparency
Low
transparenc
carrier concentration
carrier concentration
• Yablonovitch, E.; Kane, E., "Reduction of lasing threshold current density by the lowering of valence band
effective mass,
mass " Lightwave Technology,
Technology Journal of , vol.4,
vol 4 no
no.5,
5 pp
pp. 504-506
504 506, May 1986
• Yablonovitch, E.; Kane, E.O., "Band structure engineering of semiconductor lasers for optical
communications," Lightwave Technology, Journal of , vol.6, no.8, pp.1292-1299, Aug 1988
EE232 Lecture 15-2
©2008. University of California
Bernard-Duraffourg Condition in
Quantum Well
Bernard-Duraffourg Condition:
FC − FV = E − Eh1
(a) mh* > me* (as in most semiconductors)
FV > Eh1
FC Ee1
N tr = ρ e2 d ( FC − Ee1 ) =
me*
( F − Ee1 )
π = 2 Lz C
Large N tr → High threshold current
(b) mh* = me* (Ideal semiconductor)
FV = Eh1
FC = Ee1
m*
N tr = 2e
π = Lz
EE232 Lecture 15-3
∞
∫
fC ( E )dE is low
Ee1
©2008. University of California
Transparency Carrier Concentration for
Ordinary Semiconductor
(b) Ideal Semiconductor
mh* = me*
⇒
me*
N tr = 2
π = Lz
=
Transparency
p
y Condition:
FC − FV = Ee1 − Eh1
FV = Eh1
∞
∫
Ee1
1
1+ e
* ∞
e
k BTm
1
dx
2
π = Lz ∫0 1 + e x
∞
k BTme*
−x
e
=
−
ln(1
+
)
(
)
0
π = 2 Lz
=
k BTm
T e*
ln 2
π = 2 Lz
For me* =00.067
067m0
N tr ≈ 4.6 × 1017 cm −3
EE232 Lecture 15-4
E − Ee1
k BT
©2008. University of California
dE
FC = Ee1
Transparency Carrier Concentration for
Ordinary Semiconductor
(a) Ordinary Semiconductor
N tr = ρe2 d ( FC − Ee1 ) =
me*
Δ
π = 2 Lz
To estimate Δ, note that N = P
P=N e
2d
V
N=P
−Δ
k BT
−Δ
k BTmh* kBT
e
=
π = 2 Lz
⇒
e
−Δ
Δ
k BT
Δ me*
=
k BT mh*
Transparency Condition:
For mh* ≈ 6me* (in 1.55
1 55μ m laser),
laser)
FC − FV = Ee1 − Eh1
Δ = 1.43 k BT
N tr = 1.43
1 43
k BTme*
π = 2 Lz
©2008. University of California
EE232 Lecture 15-5
Effective Mass Asymmetry Penalty
N trOrdinary 1.43
=
=2
N trIdeal
ln 2
Threshold current density reduction is more than a factor of 2:
J th = J nonrad + J rad + J Auger
J th
N
= AN + BN 2 + CN 3 = + BN 2 + CN 3
qd
τ
τ : Shockley-Read-Hall nonradiative recombination lifetime
J Auger is greatly reduced when N is lowered
(1) N 3 is reduced by 8x
(2) C is also reduced due to band structure change by strain
EE232 Lecture 15-6
©2008. University of California
Bandgap-vs-Lattice Constant of
Common III-V Semiconductors
Compressive Strain
Tensile Strain
Lattice
Matched
In0.53Ga0.47As
©2008. University of California
EE232 Lecture 15-7
Qualitative Band Energy Shifts Under Strain
Biaxial
Strain
Hydrostatic
Strain
C
C
C
LH
HH
HH LH
HH,
SO
HH, LH
Hydrostatic
H
d
t ti
Strain
Biaxial
Bi
i l
Strain
C
C
δ EC
− Pε
HH, LH
SO
SO
+Qε
SO
T
Tensile
il Strain
St i
−Qε
HH
LH
SO
Compressive Strain
EE232 Lecture 15-8
©2008. University of California
Strain and Stress
ε = ε xx = ε yy
⎡σ xx ⎤ ⎡ C11 C12 C12 ⎤ ⎡ε xx ⎤
⎢σ ⎥ = ⎢C
⎥⎢ ⎥
⎢ yy ⎥ ⎢ 12 C11 C12 ⎥ ⎢ε yy ⎥
⎢⎣σ zz ⎥⎦ ⎢⎣C12 C12 C11 ⎥⎦ ⎢⎣ ε zz ⎥⎦
a − a( x)
= 0
a0
a0 : lattice constant of InP
⎧ε < 0 : compressive strain
⎨
⎩ ε > 0 : tensile strain
C
ε ⊥ = ε zz = −2 12 ε
C11
Cij : Compliance Tensor
C12 ≈ 0.5C11
Biaxial stress:
σ xx = σ yy = σ
σ zz = 0
⇒ C12ε xx + C12ε yy + C11ε zz = 0
ε zz = −2
C12
ε
C11
©2008. University of California
EE232 Lecture 15-9
Band Edge Shift
EC = Eg ( x) + δ EC
EHH = − Pε − Qε
ELH = − Pε + Qε
⎛
δ EC = aC (ε xx + ε yy + ε yy ) = 2aC ⎜1 −
C12 ⎞
⎟ε
C11 ⎠
⎝
⎛ C ⎞
Pε = −aV (ε xx + ε yy + ε yy ) = −2aV ⎜1 − 12 ⎟ ε
⎝ C11 ⎠
⎛
C ⎞
b
Qε = − (ε xx + ε yy + ε yy ) = −b ⎜1 + 2 12 ⎟ ε
2
C11 ⎠
⎝
a = aC − aV : hydrostatic potential
b
EE232 Lecture 15-10
: shear
h potential
i l
©2008. University of California
Strain Parameters in III-V
(Coldren, p.535)
EE232 Lecture 15-11
©2008. University of California
Band-Edge Profile and Subband
Dispersion
EE232 Lecture 15-12
©2008. University of California